Question: Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction.
We must first find how many positive integers less than 200 are multiples of both 18 and 24. $18=2\cdot3^2$ and $24=2^3\cdot3$, so the LCM of 18 and 24 is $2^3\cdot3^2=72$. Therefore, an integer is a multiple of both 18 and 24 if and only if it is a multiple of 72.

Dividing 200 by 72 gives quotient 2 (and remainder 56), so there are 2 multiples of 72 less than 200.

Dividing 200 by 18 gives quotient 11 (and remainder 2), so there are 11 multiples of 18 less than 200.

Dividing 200 by 24 gives quotient 8 (and remainder 8), so there are 8 multiples of 24 less than 200.

Therefore, Billy and Bobbi together can pick $11\cdot8=88$ different two-number combinations, and 2 of these involve them choosing the same number (the two multiples of 72 are the possible duplicate numbers). Thus, the probability that they selected the same number is $2/88=\boxed{\frac{1}{44}}$.